I'm not certain to which cartridge you refer. There was the old Colt .38 ACP and the later .38 Super. There was a .35 S&W auto cartridge. There's the .380 ACP. There was the .38 AMU.

Is this Smith & Wesson brand ammunition, perhaps in a blue box with white lettering?

I've fired .38 S&W factory loads in .38 Special guns over the years, at least the ones that will chamber. In a given box of factory .38 S&W ammunition I've found that some will chamber and some won't. Some can be forced with varying degrees of success. If one becomes stuck then it must be prodded out with a rod. When firing .38 S&W ammunition in a .38 Special the results are "underwhelming." Accuracy is reasonable at close range and cases eject normally. Nothing detrimental happens and nothing enriching happens either.

.38 Auto is also known as .38 Rimless Smokeless. It's a John Browning design originally chambered in the Colt 1900 "Sight Safety". The case dimensions are virtually identical to the .38 Super, but the .38 Auto is much lower pressure.

Colt 1902 Pocket (Hammer) top - .38 Rimless, Smokeless

.38 S&W is a shorter/fatter cartridge than a .38 S&W Special and the two are not interchangeable.

No boxes on either. Just marked " Peters .38 S&W" and another group "Peters .32 S&W". The ".38 auto" was a typo.......sorry. That was .32 Auto which I understand only goes in .32 semi-autos and not revolvers.